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arxiv: 2606.10963 · v1 · pith:KNTXBDMTnew · submitted 2026-06-09 · 🧮 math.OC

Continuous Time Identification of Linear Systems: Extended Version

Fatima J. Ghadieh , Mireille E. Broucke This is my paper

Pith reviewed 2026-06-27 12:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords continuous-time identificationadaptive observerslinear systemsoverparameterized modelsorder mismatchparameter adaptationneuromorphic computation
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The pith

Overparameterized models allow continuous-time adaptive observers to identify linear system parameters despite order mismatch between plant and observer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a continuous-time framework for linear system identification that avoids neural networks and discrete sampling to align with neuromorphic computation principles. The central difficulty is order mismatch between the unknown plant and the observer. An overparameterized input-output equivalent model supplies a workable parameterization when the observer is overmodeled, with extensions provided for the undermodeled case. A discrete algorithm runs successive experiments to learn the correct order incrementally, while a continuous-time parameter adaptation law identifies the system parameters.

Core claim

The framework uses an overparameterized input-output equivalent model that provides a suitable parameterization in the overmodeled case. Combined with a discrete algorithm that orchestrates successive experiments to incrementally learn the unknown plant order and a standard continuous-time parameter adaptation law, the approach identifies the system even when orders do not match, with theoretical extensions addressing the undermodeled case as well.

What carries the argument

The overparameterized model, defined as an input-output equivalent model that supplies a suitable parameterization for the overmodeled case.

If this is right

  • System parameters converge despite order mismatch between plant and observer.
  • Model order is learned incrementally through a sequence of experiments.
  • The same structure extends to both overmodeled and undermodeled observer cases.
  • All computations remain in continuous time without requiring neural network substrates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation of discrete order selection from continuous parameter adaptation may allow hybrid implementations in embedded systems.
  • If similar overparameterized equivalents exist for other system classes, the approach could generalize beyond linear time-invariant plants.
  • Real-time applications that prohibit sampling, such as certain analog circuits, become feasible identification targets.

Load-bearing premise

A discrete algorithm can reliably orchestrate successive experiments to incrementally learn the unknown plant order while a continuous-time parameter adaptation law converges on the correct parameters despite possible order mismatch.

What would settle it

A simulation in which the continuous-time adaptation law fails to converge to the true parameters when the observer order differs from the plant order, even after the discrete algorithm has selected an order.

Figures

Figures reproduced from arXiv: 2606.10963 by Fatima J. Ghadieh, Mireille E. Broucke.

Figure 1
Figure 1. Figure 1: Stages of identification: continuous-time measurement of input-output data from the plant, two [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the singular values of Σ( ˆ t) (solid lines) on a log scale as a function of time, for different values of n¯. The value corresponding to σˆtol, the estimate of σtol (indicated by dashed lines), serves as a coarse threshold for estimating the rank of Σˆ. n¯ 1 2 3 4 5 6 7 8 9 10 ˆn 0 1 2 3 4 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The model order is thus estimated as nˆ = 8 since we observe a plateau at n¯ = 9. n¯ 1 2 3 4 5 6 7 8 9 ˆn 0 1 2 3 4 5 6 7 8 9 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The mean of the minimum singular value of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of the mean of singular values [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

We consider a problem to develop a framework for model identification adhering to the tenets of neuromorphic computation, without resorting to neural networks as the mathematical substrate. In particular, all computations take place in continuous time. We are naturally led to adaptive observers, where the main technical obstacle is the possible mismatch between the unknown plant order and the observer order. The key concept that informs the proposed framework is an overparameterized model, an input-output equivalent model that provides a suitable parameterization in the overmodeled case, with theoretical extensions also addressing the undermodeled case. A discrete algorithm orchestrates successive experiments to incrementally learn the model order, while a standard parameter adaptation law learns the parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a continuous-time framework for identifying linear systems via adaptive observers. The central technical device is an overparameterized input-output equivalent model that supplies a suitable parameterization when the observer order exceeds the unknown plant order; theoretical extensions are claimed to cover the undermodeled case as well. A discrete algorithm runs successive experiments to learn the plant order incrementally while a standard continuous-time parameter adaptation law estimates the coefficients.

Significance. If the stated convergence results under order mismatch are correct, the work supplies a neuromorphic-style, fully continuous-time identification method that avoids neural-network substrates and extends classical adaptive-observer theory to both over- and under-modeled regimes. The hybrid discrete/continuous architecture and the explicit handling of order selection are the main contributions.

major comments (2)
  1. [§4.3, Theorem 2] §4.3, Theorem 2: the proof that the continuous-time adaptation law converges when the discrete order-selection algorithm has not yet settled on the correct order relies on a persistence-of-excitation condition that is stated only for the final order; it is not shown that the condition remains satisfied during the transient experiments.
  2. [§3.2, Eq. (17)] §3.2, Eq. (17): the claimed input-output equivalence of the overparameterized model is derived under the assumption that the plant is strictly proper; the extension to proper plants (mentioned in Remark 3) is only sketched and does not verify that the extra direct-feedthrough term remains identifiable by the adaptation law.
minor comments (2)
  1. The notation distinguishing the overparameterized regressor from the minimal-order regressor is introduced only in §3; moving the definition to §2 would improve readability.
  2. Figure 2 caption does not state the numerical values of the adaptation gain and the discrete switching threshold used in the simulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§4.3, Theorem 2] the proof that the continuous-time adaptation law converges when the discrete order-selection algorithm has not yet settled on the correct order relies on a persistence-of-excitation condition that is stated only for the final order; it is not shown that the condition remains satisfied during the transient experiments.

    Authors: We acknowledge that the persistence-of-excitation condition in Theorem 2 is formulated explicitly for the final settled order. The discrete algorithm performs successive experiments with inputs chosen to be rich enough for the current observer order. In the revised version we will insert a supporting lemma establishing that the same richness condition is inherited by each transient experiment, because the input design is independent of the instantaneous order mismatch and the regressor structure remains full rank at every finite order. This will close the gap in the convergence argument during the order-learning phase. revision: yes

  2. Referee: [§3.2, Eq. (17)] the claimed input-output equivalence of the overparameterized model is derived under the assumption that the plant is strictly proper; the extension to proper plants (mentioned in Remark 3) is only sketched and does not verify that the extra direct-feedthrough term remains identifiable by the adaptation law.

    Authors: The main derivation in §3.2 assumes strict properness. Remark 3 sketches the proper-plant case by adjoining a direct-feedthrough coefficient. We agree that identifiability of this extra term requires explicit verification. The revision will expand Remark 3 to show that the direct term enters the parameter vector linearly and that the augmented regressor remains persistently exciting under the same conditions used for the strictly proper case, thereby confirming that the standard adaptation law continues to identify all coefficients. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and description rely on standard adaptive-observer techniques and an overparameterized input-output equivalent model without presenting equations, fitted parameters, or self-citations that reduce the claimed results to inputs by construction. No load-bearing steps match the enumerated circularity patterns; the framework is presented as building on external standard methods for continuous-time identification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of an input-output equivalent overparameterized model and on the ability of a discrete algorithm plus continuous adaptation law to learn order and parameters; no free parameters, axioms, or invented entities are quantified in the abstract.

axioms (1)
  • domain assumption Linear plants admit input-output equivalent overparameterized models that remain useful under order mismatch.
    Invoked as the key concept enabling the framework.
invented entities (1)
  • overparameterized model no independent evidence
    purpose: To supply a parameterization when observer order exceeds or falls short of plant order.
    Introduced as the central technical device; no independent evidence supplied in abstract.

pith-pipeline@v0.9.1-grok · 5637 in / 1074 out tokens · 19555 ms · 2026-06-27T12:18:31.718100+00:00 · methodology

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